This video continues at kzread.info/dash/bejne/i5xls7eSnpXMY6w.html and delves into so-called Untouchable Numbers. More Ben Sparks on Numberphile: bit.ly/Sparks_Playlist

@BlockdaCoolguy

Ай бұрын

Do 5

@PhilBagels

Ай бұрын

Even though I know about number theory, and know about perfect, abundant, deficient, amicable, sociable, I had never heard of aspiring numbers before. Now because we have names for all of these categories, we seem to need one more. Doing the aliquot process once, divides all numbers into three categories: deficient, abundant, and perfect. But doing an aliquot sequence, we get (potentially) seven categories, but three of them don't seem to have names: Perfect - stay the same forever. Aspiring - eventually get to a perfect number. Amicable - bounce back and forth between two values. Sociable - cycle through a loop of more than two numbers. ?1? - the ones that never get to a loop or perfect number - there might not be any in this category. ?2? - numbers that eventually get to a loop. You might say they "aspire to be amicable or sociable, rather than aspiring to be perfect". ?3? - the numbers that get to 1 eventually. Note that both abundant and deficient numbers can fall into this category. I guess those ?1? numbers, if they are found to exist, can be named after whoever finally proves their existence. The ?2? numbers could be called "shy" numbers - they're trying to get into the amicable/sociable group. I suppose this category could be split into two. And the ?3? category in which the majority of numbers fall, should have some name, too. At first, I was thinking to propose calling them "mortal" numbers, because through the aliquot sequence, they eventually "die". But that seems too dark of a name.

@CheckmateSurvivor

Ай бұрын

The next puzzle for you to solve: The 300 Coins Problem. 300 coins are placed randomly on a table. A 300 letters long message (Signal) is written, one letter per coin, that would lead to a hidden treasure. Then the coins are flipped over and a randomly generated Noise 300 letters long is written on the other side of coins. The coins then get put in a bag and scrambled. Finally, the coins are put back on the table. Your task is to flip and move the coins around until the original message is recreated. Can you do it?

@Einyen

Ай бұрын

I checked wikipedia on sociable numbers for my own curiosity, and if it is accurate then: The only known loop lengths are 1 (perfect), 2 (amicable), 4, 5, 6, 8, 9 and 28. (and 5, 9 and 28 only have 1 known sequence each) "It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n." So loops with length n=4k+3: 3,7,11,15,... is probably/maybe not possible.

@ZeraAuraeditz

Ай бұрын

U still exist?

The Numberphile Conjecture: If you give numberphile enough time, every integer will have a video about it.

@numberphile

Ай бұрын

That's the plan

@guillermojperea6355

Ай бұрын

Absolutely beautiful and simple conjecture! And i love that that's the plan!

@alveolate

Ай бұрын

the numberphile playlist of all videos will then become an OEIS sequence since it will have a unique sequence of integers by age of video.

@thewhitefalcon8539

Ай бұрын

Fun fact: 9538 is the smallest number that can't be defined in 30 English words or less.

@NStripleseven

Ай бұрын

@@thewhitefalcon8539”Nine thousand five hundred thirty-eight”

296 🤦‍♀ (my wife is now not speaking to me for 284 days apparently)

@NorlanderGT

Ай бұрын

Was it just a brainfart, or did you think about 296 for different reasons and got it mixed up?

@SparksMaths

Ай бұрын

I think I had 496 in my head (for perfect reasons) and it contaminated my thoughts. Mea culpa. 🫤

@d4slaimless

Ай бұрын

Epic fail )

@camileonico

Ай бұрын

🫂

@DadgeCity

Ай бұрын

@@NorlanderGT the answer is at 4:02

The fact that he doesn't know the number that's on his wife's half of the heart is concerningly humorous

@cheweh842

Ай бұрын

something something keychain parties

@CWinterstorm

Ай бұрын

I think he's ending up in the dog house for a while ;)

@c.jishnu378

Ай бұрын

Time stamp?

@soyokou.2810

Ай бұрын

4:58

@lyrimetacurl0

Ай бұрын

284

brady commentating the 138 graph has me hysterical oh my lord

@soyezegaming

Ай бұрын

Here before this comment is popular

@camileonico

Ай бұрын

masterpiece

@hamc9477

Ай бұрын

It was the "go son!!" That sent me

@stuiesmb

Ай бұрын

They need him in as a guest commentator on @jellesmarbleruns

@simonf8370

Ай бұрын

Made my day and it's not even 8am!

8:58 "It's so over!" 9:01 "We're so back!" 9:04 "It's so over!" 9:12 "We're so back!"

@daemoneko

Ай бұрын

in the midst of "its so over", I found there was within me, an invincible "we're so back!"

@Aravaganthus

Ай бұрын

I looked specifically for this comment

@RazvanMihaeanu

Ай бұрын

Brought to you by... Jelle's Marble Runs!

@sergio_henrique

Ай бұрын

Reminds me of Tetris gameplay shooting for some crazy world record breakthrough.

@andrewwang2209

Ай бұрын

WHEEEEEEE

This is really what numberphile is all about

@lvdovicvs

Ай бұрын

This is the video I'm going to cite for the foreseeable future when someone asks what number theory is. And I'm going to foist it on my kids tonight

@stephenbeck7222

Ай бұрын

Just need Tadashi Tokeida to incorporate some weird toy into it

@akaelalias4478

Ай бұрын

+

That amicable number heart keychain is one of the nerdiest romantic thing I've ever heard of - it's very cute

@HasekuraIsuna

Ай бұрын

Didn't James Grime mention this as a thing to do when he taught us amicable numbers like 10 years ago?

@soyezegaming

Ай бұрын

Here before this comment is popular

@abydosianchulac2

Ай бұрын

​@@HasekuraIsuna I wonder if that's where Ben got the idea from.

@gladiatorsfc7

Ай бұрын

So romantic to forget your wife's number

@philipwilson46

Ай бұрын

You can buy the keyrings at Maths Gear.

This is why I love mathmatics: a relatively simple question leads to a whole mini world of calculations and mysteries.

@daniel_77.

Ай бұрын

The universe doesn't care about intuition 😂

@vikashchandra9917

Ай бұрын

@@daniel_77.your comment makes no sense

@daniel_77.

Ай бұрын

@@vikashchandra9917 Sorry. I meant that the things we see and do, even the seemingly simple natural numbers, still hides a lot of complex reasoning. When things may seems obvious and intuitive, In reality it doesn't work like that.

He's gonna have to sleep on the couch tonight because he forgot his wife's amicable number... AGAIN!

@ljfaag

7 күн бұрын

It should be easy enough to recalculate if you forgot shouldn't it

Brady's commentary of the highs and lows of 138 was awesome

I like to imagine that 276 goes all the way up straight to the first and only odd perfect number, and that number also happens to be the first number to start a loop that disproves the collatz conjecture.

@Lucashallal

Ай бұрын

Lol that would be funny

@byronrobbins8834

Ай бұрын

​@@Lucashallaltrying to square each digit of 276, and add them together, you will then arrive into the melancoil, which is problematic.

Just want to point out that the first number that does a really wild ride was 138, and the next number he showed was 276, which is exactly double. And then the next Lehmer five is 552, again exactly double.

@AndyWitmyer

Ай бұрын

One wonders if 276 and 552's aliquot trajectory would ultimately be in some way analogous to 138's, except by several orders of magnitude longer in sequence

@smeejay9621

Ай бұрын

If you look at it in terms of using 138 as a base number n, 3 of the 5 numbers are multiples of n. 2n, 4n, 7n.

@gordontaylor2815

Ай бұрын

@@AndyWitmyer 276 already has an index (sequence length) over 2,100 and 564 has an index near 3,500 currently. 138 ONLY took an index of 177 to resolve, thus both sequences are already at least one order of magnitude larger and show no signs of ending anytime soon.

@patrickmckinley8739

Ай бұрын

So lets see how much magic 276 has in it. 3 times 276 is 828. The sequence is open. It joins the 660 sequence immediately. Also one of the Lehmer Five.

@byronrobbins8834

Ай бұрын

​@@patrickmckinley8739you ought to try something like, 4+49+36, which becomes the number 89, and this point, we land in the melancoil.

11:16 "The answer is... We don't know" Brady, utterly disappointed: "Of course not..."

@randomname285

Ай бұрын

you mathematicians don't know shi...

8:46 What an absolute roller coaster ride of emotions!

@numberphile

Ай бұрын

I'm still recovering

@dielaughing73

Ай бұрын

​@@numberphile I think we should call them "rollercoaster numbers"

The best part of the video is where he watches the Price of Bitcoin

@ShaunakDesaiPiano

Ай бұрын

I was about to say!… the path for 138 looks like a stock price.

@handsome_man69

16 күн бұрын

I am a handsome man

For the rest of his days, Ben is going to wake in a cold sweat remembering the time he got 296 wrong. If his friend group is anything like mine, they'd never miss an opportunity to bring it up.

@NEMesis1413

Ай бұрын

It'll be his version of the Parker square

This feels like the 3n+1 conjecture, but finding an actual number that blows to infinity!

@rtpoe

Ай бұрын

You noticed that, too!

@RetardedSissy

Ай бұрын

It practically is, in more ways than one.

@laxrulz7

Ай бұрын

This was my thought. If you want to really chase a rabbit hole google Muratz Conjecture in relation to Collatz and you start to see real similarities. Wonder if there's something there.

@GWaters-xr1fv

Ай бұрын

You mean : the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1. Yes, a very similar situation that also came to my mind ( and many others I'm sure ). Great video Ben !

An instant classic! Great job guys

@numberphile

Ай бұрын

Cheers - glad you enjoyed it

The first time I programmed this was in the 80s on a C64. I hit brick walls several times; first my algorithm to compute the proper divisor sum was too simple and thus too slow for the gigantic numbers I ran into for the 138. When I fixed that, they still kept growing beyond the numbers the programming language could handle. I had to restart the whole programming several times until I found what I really was looking for: These things which I now just learned are called sociable loops. I called them circles. Later I found them again in the OEISⓇ. Very nice to see all my steps again in this video now. ☺

220 and 296. The Parker Heart.

@jj.wahlberg

Ай бұрын

HAHA

@smicksatusadotnet

Ай бұрын

The Sparks Amicable

@4thalt

29 күн бұрын

Parker: doing something that's almost right, but just wrong enough so it doesn't work Sparks: doing something wrong confidently, but knowing the right answer

If you ever doubted yourself after all these years Brady - you still got it. Absolute banger of a Numberphile video!

5:27 it was almost physical the amount of relief I felt seeing the correct number on the other half of the heart. ❤️

Brady cheering on 138 is so funny.

@tonybates7870

Ай бұрын

GO ON SON!!!

When I was 17 I saw James Grime’s video on amicable numbers and he showed us the keychains with 220 and 284. Being the nerdy 17-year-old I was, I bought them. I held onto those for about 6 years, until I finally had a long-term boyfriend to give one of them to. He’s an engineer so not quite as into pure math as I am, but he’s quite a good sport about his 220 wooden heart.

@JohnSmith-nx7zj

18 күн бұрын

You don’t often read a story so wholesome and heartwarming on the internet.

In chemistry, an aliquot is taking off part of your solution and then only doing something with that part rather than the whole solution.

8:49 I guess I'm a Nerd, I was genuinely excited & cheering the number on as it went. lol

I expected the answer to “are there any sequences that don’t collapse?” to be “we don’t know”. Especially since they’d already said it was a conjecture. But I’d never had guessed the first candidate would be such a low number unlike with the Collatz conjecture.

@AsterothPrime

Ай бұрын

True, although the number 27 in the Collatz conjecture is a low number, yet blows all the way up to 9232 in a similarly shocking manner, but not quite like this! This is a more fundamental number theory, of which the Collatz conjecture is a more complex flavour.

5:30 Whoops, that's worth at least an extra flower in the next bouquet.

This is the best Numberphile that I've seen in years

You'd think to find things like this you need to invent something complicated. But here we have very easy algorithm that suddenly blows out and away so we don't even have enough computational power to check the end result. Loved the video!

@JohnSmith-nx7zj

Ай бұрын

There’s a lot of thing like that that amaze me. It’s trivial to prove that if you gather 6 people together, either you have 3 mutual acquaintances or 3 mutual strangers. 18 ensures 4 mutual acquaintances or strangers. But the minimum number to ensure 5 mutual acquaintances or 5 mutual strangers is still unknown (except that it’s between 43 and 48).

This was an awesome sort of "back to the roots of Numberphile" video, and the general excitement overall from both Ben and Brady were just great.

The 138 moment is how i feel about every sequence. Get kind of familiar with the general characteristics of the sequence, and then get blown away by a result.

some of these numberphile videos genuinely shock me to my core well done

For those interested, aliquot sequnce for 276 is currently at step 2146, not 2090. The last advance was made in January 2024, when a C209 was split into a P98 and P112. That means the number of digits, C for composite, P for prime. C209 is the supercomputer (or rather a distributed computing project) territory with months/years of GNFS sieving required to factor it. The previous hurdle was step 2140, passed in August 2022 after factoring a C213 which turned out to be P97 * P116.

"Of course." - I will never regret subbing to your channel.

I was sure this was going to be one of those situations like Collatz, where we're sure that everything goes to zero and it's just annoyingly difficult to prove... so it came as a big surprise, even knowing the title of the video, that we have a specific low number that we think might actually be a counterexample!

@JohnSmith-nx7zj

Ай бұрын

Yeah I was shocked how low the first number is where we haven’t figured out the answer.

@TimSorbera

Ай бұрын

I think this is like Collatz, technically we don't know but (having spent a lot of time with these sequences) my suspicion is that infinity is an awfully long time for it to *not* end at some point. I think they will all end, it just takes enormous amounts of computations to check

@JohnSmith-nx7zj

Ай бұрын

@@TimSorbera the difference with collatz is that we know the answer for all starting numbers up to something like 2^60. It’s wild to me that we don’t know the answer for a starting number as low as 276.

@JohnSmith-nx7zj

Ай бұрын

@@TimSorbera Richard K Guy presented some evidence for a counter-conjecture that there are unbounded aliquot sequences.

I love all the Numberphile alumni but I always come back to Ben. Top 10 Numberphile videos are probably 40% Ben Sparks here.

"Are there any that don't come back." My immediate thought was, "It's a conjecture--we don't know."

The Australian accent is perfect for providing passionate commentary on an evolving graph!

"LIKE MARBLE RACING" I LOVE THIS MAN

These are my favorite numberphile videos. Great stuff

@byronrobbins8834

28 күн бұрын

You ought to test the number 276 out for happiness and perfection.

Excellent video! Original Numberphile :D

@numberphile

Ай бұрын

O.G.

I loved the commentary for 138 :D It gave me a really good laugh! And also the youtube channel idea hahaha, brilliant

Never this channel fail to amaze me. This is one of those that are so simple to understand that is mind blowing

Videos about a specific number like this are the best

Always a pleasure to see Ben. I was struggling with GCSE maths when he became my teacher and I went on to get a Masters Degree in Physics - one of the best teachers I have ever had

I love Ben's videos. Also he looks different than the previous video whenever he has been gone for awhile.

I miss these old kind of videos! And Ben is always a treat!

@Pathakin.

Ай бұрын

Threat?

@HasekuraIsuna

Ай бұрын

@@Pathakin. Gotta love autocorrect 🙃

I LOVE THIS ONE! it's so exciting! this might be the first numberphile video that made me laugh out loud with joy and excitement. Also kudos to Ben; he's been responsible for several of my favorite numberphile videos.

8:47 is one of the most satisfying rides in numberphile history ❤

Perhaps my favorite Numberphile to date!

Love the old school style videos, love Ben's enthusiasm, great video for my sunday morning, thanks lads

I can't believe y'all is still coming up with videos like this are all these years. You're legends

One of the most exciting and touching video i've seen on KZread. thanks again Numberphile

I frickin' love Ben

Fabulous video! Always a mindblowing experience watching Numberphile videos! This one was particularly inspiring 🙏🙏🙏

This is a very cool one, exactly the kind of stuff that interests me. Ben is so good at explaining this kind of stuff

For those new to the topic - you can check the known factorizations for any sequence on factordb

cool! a new video from Numberphile yay!!!!!

Absolutely do that, that was a fun sequence to watch with the commentary

This feels like just the collatz conjecture with extra steps! :D

I'm surprised this doesn't attract more attention, if only because it would imply there are trajectories that can flawlessly avoid primes without being a trivial sequence of multiples. If there are numbers that trend to infinity, then the patterns they follow would be another insight into the patterns that primes follow

@gordontaylor2815

Ай бұрын

Those doing research on these sequences have noticed a few patterns (the technical term is "guides") generally based on two principles: * How many powers of two the number you're looking at has (fewer means smaller numbers and more means larger numbers) * Is there any power of three in the number (if yes -> bigger numbers, if no -> smaller numbers) You generally want the terms in the sequence getting smaller because that increases your odds of it terminating by hitting a prime (or some kind of cycle of numbers).

i love this channel so much

@numberphile

Ай бұрын

We love the people who watch it!

I love a classic numberphile "number " video ! Hope to see a lot more of them !

I am a simple beast. I see a numberphile thumbnail that's just a number written across the screen, and I click it

absolutely loved this oldschool type of video

Great video, thanks!

Might be my favourite Numberphile video yet. Simple, pure maths that an 8-year-old can understand, but with a deep complexity that leaves the greatest mathematicians clueless. The content of this video is more universal than the Universe. It existed before the Big Bang, and will still exist after the Big Crunch. Perfect.

Next video better be Ben explaining why we haven't found an odd perfect number

@gordontaylor2815

Ай бұрын

Part of the problem is that if odd perfect numbers DO exist (many people think they don't) they're going to be very large numbers to work with - the current best estimate of the smallest one is at LEAST 2300 digits with 48 factors!

@byronrobbins8834

27 күн бұрын

​@@gordontaylor2815we might also test several numbers for happiness, as to whether you get to the number 1, or approach the melancoil.

Classic Numberphile!!!!!! ❤❤❤

@deliciousrose

Ай бұрын

Write phyton code to check ❌️ Write code in geogebra ✅️

I spent a few years factoring aliquot sequences with my computer in its spare time. It can be a lot of fun to see the sequences progress and learn the math of the ups and downs as well as the factoring algorithms and tools.

Love this... particularly the animations. Is @SparksMaths going to do a live build video for the GeoGebra applet that he used? I hope so! Ben and Brady, thanks for another great video!

@dielaughing73

Ай бұрын

Link to the file is in the description, in case you missed it

I squealed with glee when Ben's face popped up. I love his communication skills and his topics.

Appreciate ya. Thanks for sharing.

Stunning!

Simply amazing.

Love this!!!!

Love Ben!

"Maybe it's a perfect number?" "It's an aspiring number" haha i love it

Always interesting!

Brady still amazing with his genuinely excitement ❤🎉

Plot on log scale!! Edit: oh, thanks. Whew!

I'm all aboard for Number Racing channel! Is that what we're calling it? Number Racing?

A fascinating video

5:26 Numberphile is always answering the really important questions.

Ben Sparks 🤝 Geogebra files

So is so interesting to watch 😮

After many years, Ben still doesn't know the key number to his wife's heart? 🤔

This is fascinating! 10:15 I would definitely visit that website.

Thoroughly engaging

I'm so happy that Brady watches marble racing!

Very cool!

I love this

276 / 4 = 69. Boom I just made it even more interesting

@randomname285

Ай бұрын

so 8 person mutual ...... leads to infinity?

@dielaughing73

Ай бұрын

Obligatory "Nice."

Yay!!! Old school numberphile!

Bro is one digit away from summoning a fandom

@emperortgp2424

Ай бұрын

what fandom

@MathNerd1729

Ай бұрын

​@@emperortgp2424 Based on the account, I assume they're referring to the number 2763 being mentioned multiple times in Battle For Dream Island episodes. Hope that helps! :)

@PlatonicPluto

Ай бұрын

The prophecy is spoken, we must test it.

@zaydenmYT

Ай бұрын

@@MathNerd1729 yes

138 is really a a rollercoaster in aliquot sequence

Wow. I was thinking that it wouldn't be amazing. But, it was!

Superb video

@numberphile

Ай бұрын

Thanks